where \(G_F\)  denotes the Fermi constant, the fine structure constant \(\alpha _e=\frac{e^2}{4\pi }\) , the chiral projection operators \(P_{L,R}=(1\mp \gamma _5)/2\) , \(\sigma _{\mu \nu }=\frac{i[\gamma _\mu ,\gamma _\nu ]}{2}\) , \(\lambda _{q_1q_2}\)  denotes the Cabibbo-Kobayashi-Maskawa (CKM) elements, and \(C_{i}\)  denote Wilson coefficients.
For the \(b\rightarrow s/d\ell ^+\ell ^-\)  transitions via the \(u\bar{u},c\bar{c}\)  loops, \(\lambda _{bs(bd)}=V_{tb}V^*_{ts}\) (\(V_{tb}V^*_{td}\) ), and the expressions of Wilson coefficients \(C^{eff}_{7,9}\)  and \(C_{10}\)  are given in Ref. [1]}. For the \(c\rightarrow u\ell ^+\ell ^-\)  transition via the \(d\bar{d},s\bar{s}\)  loops, \(\lambda _{cu}C^{eff}_{7,9}=\frac{4\pi }{\alpha _s}\Big [V^*_{cd}V_{ud}C^{eff(d)}_{7,9}(q^2)+V^*_{cs}V_{us}C^{eff(s)}_{7,9}(q^2)\Big ]\)  as well as \(C_{10}=0\)  and the expressions of \(C^{eff(s,d)}_{7,9}(q^2)\)  can be found in Refs. [2]}, [3]}. For the \(s\rightarrow d\ell ^+\ell ^-\)  transition via the \(u\bar{u}\)  loop, \(\lambda _{sd}=V_{us}V^*_{ud}\) , \(C^{eff}_7\approx \frac{V_{cs}V^*_{cd}}{2V_{us}V^*_{ud}}c^c_{7\gamma }\) , \(C^{eff}_9=(z_{7V}-\frac{V_{ts}V^*_{td}}{V_{us}V^*_{ud}} y_{7V})\frac{2\pi }{\alpha _e}\) , and \(C_{10}=-\frac{V_{ts}V^*_{td}}{V_{us}V^*_{ud}}\frac{2\pi }{\alpha _e}y_{7A}\)  with \(z_{7V}=-0.046\alpha _e\) , \(y_{7V}=0.735\alpha _e\) , \(y_{7A}=-0.700\alpha _e\)  as well as \(c^c_{7\gamma }=0.13\alpha _e\)  from Refs. [1]}, [5]}, [6]}.
For \(T_{b3}\rightarrow T_8\ell ^+\ell ^-\)  and \(T_{c3}\rightarrow T_8\ell ^+\ell ^-\)  decays, the Wilson coefficient \(C^{eff}_9\)  receives not only from the four quark operators but also from the long distance (LD) contributions coming from \(c\bar{c}\)  for \(T_{b3}\rightarrow T_8\ell ^+\ell ^-\)  and \(d\bar{d},s\bar{s}\)  for \(T_{c3}\rightarrow T_8\ell ^+\ell ^-\) . Note that the \(T_{c3}\rightarrow T_8\ell ^+\ell ^-\)  decays are
dominated by LD contributions. For \(T_{8}\rightarrow T^{\prime }_8\ell ^+\ell ^-\)  decays, the LD contribution arises mainly from the photon-mediated process \(T_8\rightarrow T^{\prime }_8\gamma ^*\rightarrow T^{\prime }_8\ell ^+\ell ^-\)  [7]}, [8]}.
